The teacher of mathematics should engage in ongoing analysis of teaching and learning by-

observing, listening to, and gathering other information about students to assess what they are learning;

examining effects of the task, discourse, and learning environment on students' mathematical knowledge, skills, and dispositions;

in order to-

ensure that every student is learning sound and significant mathematics and is developing a positive disposition toward mathematics;

challenge and extend students' ideas;

adapt or change activities while teaching

make plans, both short- and long-range;

describe and comment on each student's learning to parents and administrators, as well as to the students themselves.

Elaboration

Assessment of students and analysis of instruction are fundamentally interconnected. Mathematics teachers should monitor students' learning on an ongoing basis in order to assess and adjust their teaching. Observing and listening to students during class can help teachers, on the spot, tailor their questions or tasks to provoke and extend students' thinking and understanding. Teachers must also use information about what students are understanding to revise and adapt their short- and long-range plans: for the tasks they select and for the approaches they choose to orchestrate the classroom discourse. Similarly, students' understandings and dispositions should guide teachers in shaping and reshaping the learning environment of the classroom. Additionally, teachers have the responsibility of describing and commenting on studentsâ learning to administrators, to parents, and to the students themselves.

Studentsâ mathematical power depends on a varied set of understanding, skills, and dispositions. Teachers must attend to the broad array of dimensions that contribute to students' mathematical competence as outlines in the Curriculum and Evaluation Standards for School Mathematics. They should assess students' understandings of concepts and procedures, including the connections they make among various concepts and procedures. Teachers must also assess the development of studentsâ ability to reason mathematically-to make conjectures, to justify and revise claims on the basis of mathematical evidence, and to analyze and solve problems. Students' dispositions toward mathematics-their confidence, interest, enjoyment, and perseverance-are yet another key dimension that teachers should monitor.

Paper-and-pencil tests, although one useful medium for judging some aspects of students' mathematical knowledge, cannot suffice to provide teachers with the insights they need about their students' understandings in order to make instruction as effectively responsive as possible. Teachers need information gathered in a variety of ways and using a range of sources. Observing students participating in a small-group discussion may contribute valuable insights related to their abilities to communicate mathematically. Interviews with individual students will complement that information and also provide information about students' conceptual and procedural understanding. Students' journals are yet another source that can help teachers appraise their students' development. Teachers can also learn a great deal from closely watching and listening to students during whole-group discussions.

As they monitor students' understandings of, and dispositions toward, mathematics, teachers should ask themselves questions about the nature of the learning environment they have created, of the tasks they have been using, and of the kind of discourse they have been fostering. They should seek to understand the links between these and what is happening with their students. If, for example, students are having trouble understanding inverse functions, is it because of the kinds of tasks in which they have been engaged? Is it related to the ways in which the group has explored and discussed ideas about functions and their inverses? Although it may be that the students lack prerequisite understandings, it could also be that this is a difficult piece of mathematics or that the teacher needs to consider alternative ways to help students "unpack" the ideas. Or, if students quickly give up when a direct route for solving a problem is not apparent, teachers must consider how the experiences that students have been having and the environment in which they have been working may not have helped them to develop the perseverance and confidence they need. Teachers need to analyze continually what they are seeing and hearing and explore alternative interpretations of that information. They need to consider what such insights suggest about how the environment, tasks, and discourse could be enhanced, revised, or adapted in order to help students learn.

Vignettes

6.1 Some teachers begin to change by allocating one day a week to "different" mathematics activities. Although the Curriculum and Evaluation Standards makes clear that the goal is for problem solving, reasoning, and communication to be interwoven throughout the curriculum, teachers must experiment with alternative approaches to changing their practice. This "one day a week" strategy is one such approach-not the goal, but for some teachers, a viable first step.

Ms. Levesque has been having students working in groups of four on Fridays, solving nonroutine problems. Last week, she had them work on the handshake problem. (If ten people are at a party and everyone shakes everyone else's hand exactly once, how many handshakes take place?) Things seem to be going quite well. The students appear to enjoy these Friday sessions and she looks forward to them herself.

She notices, however, that when she listens to students talking among themselves before class, they still groan about the word problems on their daily homework. Many students leave this part of their work unfinished because, they say, the problems are too hard.

As Ms. Levesque compares what she is learning about her students with her goals for them, she is troubled, because she wants her students to feel confident about solving mathematical problems and to stick to them even when the problems are hard.

The teacher analyzes what she has been doing-the tasks she has been using and the environment and discourse she has created around them-and comes up with an alternative plan to try, based on her analysis of the situation.

Talking and collaborating with colleagues can enhance the analysis and the process of improving instruction.

She thinks about those Friday sessions. Why aren't they fostering these dispositions toward mathematical problem solving? Ms. Levesque wonders whether perhaps these special sessions seem to the students to be separate from "real math." It is, after all, just one day a week-and what they do on the other four days is quite different in spirit and in content.

Ms. Levesque decides to try working on word problems together for part of the period every day for a while to see if that makes a difference. She will try having them discuss the problems, examining different approaches and solutions, instead of just going over the answers together. In addition, the students will keep a journal or notebook in which to record strategies and reflections.When she talks to her department head about this over lunch, her department head says that she has had a similar concern with her classes and that she, too, will try Ms. Levesque's plan and they can compare notes after a few weeks.

The teacher gathers information about what students have learned.

Instead of relying on his assumptions to explain what he has found, the teacher decides to gather some more information that might help him understand what has gone wrong. He assumes that there may be reasons for the childrenâs performance that go beyond carelessness.

6.2 The second graders have just finished working on addition and subtraction with regrouping. On a written test, many of them "forget" to regroup when they need to in subtraction. Instead, they do this:

The teacher, Mr. Lewis, thinks they are being careless. He feels a little annoyed because this is something on which he has spent a lot of time. He decides, though, that he should sit down with the children one by one for a few minutes and have them talk through a couple of the problems and how they solved them. He thinks he may be able to tell what they are doing wrong this way.

He chooses a couple of problems from the test and asks the children to justify their answers using bundles of Popsicle sticks. He discovers that most of them are not connecting the work they did in class with manipulatives to these written problems. When they have the Popsicle sticks, they find that their answers don't make sense, and they revise them to match what they do with the sticks.

Mr. Lewis had assumed that if they "saw" the concepts by actually touching the objects, they would understand. He now thinks that maybe he didn't do enough to help them build the links between the concrete model and the algorithm. He starts wondering what he could do to help them make that connection better. He hypothesizes that maybe they know how to regroup but may not understand why or when regrouping is necessary. He decides to make up a worksheet with examples where regrouping is necessary and some where it is not and have the children discuss whether or not they would have to regroup in each case and how they know that.

6.3 Ms. Lundgren has been trying to change her approach to teaching mathematics so that students are learning to reason and communicate about mathematics, to make sense of mathematical ideas, and to make connections. She believes she has been successful in moving the discourse of her classroom away from a focus on right answers and the teacher as authority.

The teacher knows that she must find some ways of documenting and assessing what students are learning, especially in view of her new goals for them. She finds it helpful to work with a colleague.

The teacher wants the parent to understand both what her child is doing and what is being held as important in her mathematics class.

Because it enables her to give the parent specific examples, her system of cards as an index to the childrenâs journals helps her to do both.

The teacher got this idea from the NCTM Curriculum and Evaluation Standards for School Mathematics, (pp. 235,236). She and her colleague found several ideas there for assessing and keeping track of studentsâ learning.

The teacher has a systematic way of collecting and analyzing information about her own teaching.

Although she finds it difficult, she has also been devising better mathematical tasks, she thinks. With the help of the other fifth-grade teacher, Ms. Lundgren has also come up with some ways of keeping track of what students are learning. Today she is meeting with parents to go over their children's report cards, and she has decided to draw on her new records for these conferences.

When she is meeting with Mrs. Byers, Stacy's mother, Ms. Lundgren wants to show her how Stacy is making connections in division. Looking at her card on Stacy, Ms. Lundgren tells the mother that Stacy was able to explain how, for 288, 3 r 4 was the same answer as 3.5 (a quotient obtained on the calculator) but also how the two answers differed. Ms. Lundgren, having made a note of it, opened Stacy's mathematics journal to the page where Stacy had worked this out. Then, referring to the index card again, Ms. Lundgren shows Stacy's mother all the ways that Stacy found to represent 8 in her journal. Because she also wants to talk with Mrs. Byers about Stacy's disposition toward mathematics, Ms. Lundgren refers to a chart she is keeping on her students' mathematical attitudes. With this chart, she has periodically made notes to herself. She has also had her colleague next door come in and observe once a month and make notes on the chart for her. Mrs. Byers finds all these specific examples very useful and comments that she thinks that what Ms. Lundgren is trying to do in math is great and she wishes she had had a mathematics class like this when she was in school.

6.4 Ms. Weissmann has been audiotaping her mathematics classes each day this year. She listens to as much of each tape as possible while she plans for the next day's class. In listening to herself and to the students, she begins to notice a pattern.

This pattern is not uncommon, but it is troubling to this teacher, who has always been interested in, and relatively successful with, mathematics. She also is convinced that things do not have to be like this.

The teacher selects some simple ways of maintaining a record of what is going on in her class.

On the one hand, the girls are very quiet and speak softly and say "I don't know" at least as often as they say anything. The boys, on the other hand, are loud, and she hears herself calling them by name a lot. They participate actively in the mathematics discussions as well as in their own little games and fooling around. She begins tallying the frequency with which she calls on boys and on girls. She also begins a chart for what the boys and girls each contribute to class discussions, not just how often.

At the same time, Ms. Weissmann gets a couple of books from the library, both centered on discourse and on women's patterns of interaction in different settings. She decides to make this a project for herself: to improve the balance of kinds and frequency of participation among boys and girls in the class discussion. She also plans to be alert if there are other such patterns underlying the boy-girl split.

The teacherâs "project" helps her to focus on an issue that is of great importance to her.

Summary; Analysis

Analysis of instruction recognizes the intimate relationship between teaching and assessment. To improve their mathematics instruction, teachers must constantly analyze what they and their students are doing and how that is affecting what the students are learning. Using a variety of strategies, teachers must continuously monitor students' capacity and inclination to analyze situations, frame and solve problems, and make sense of mathematical concepts and procedures. Teachers should use such information about students to assess not just how students are doing, but also to appraise how well the tasks, discourse, and environment are working together to foster students' mathematical power and to adapt their instruction in response.